Let $\{\phi_i\}$ the sequence of the eigenfunctions of laplacian operator on a domain $\Omega$, that is, considering $\{\lambda_i\}$ the respective eigenvalues, we have
$$
\int \nabla \phi_{i} \cdot \nabla \psi = \lambda \int \phi_{i} \psi, \forall \psi \in H^{1}_{0}(\Omega).
$$ Define the space
$$
V=\text{span}\{\phi_1,..., \phi_k \}.
$$
Is it true that the uniform norm is well definido on $V$? That is,
$$
||u||_{\infty} = \sup_{\Omega}{|u(x)|} < \infty, \forall u \in V?
$$
The context: My professor said that the norm $||\cdot||_{\infty}$ and $||\cdot||_{H^{1}_{0}}$ are equivalent in $V$ because it has finite dimension. I couldn't use any Sobolev embedding theorem to conclude that $H^{1}_0(\Omega)$ is contained in $L^{\infty}(\Omega)$ because all I know its $H^{1}_0(\Omega)$ is continuously embedded in $L^{q}(\Omega)$ for all $q \in [1, 2^*]$.